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Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \setlength\paperheight {10in}% \setlength\paperwidth {13.5in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Towards the cone conjecture\\ for hyperk\"ahler manifolds} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf Simons Center \\ Quiver Varieties Program Seminar\\ November 8, 2013 } \end{center} \newpage {\bf \blue The K\"ahler cone and its faces} \def\Teich{\operatorname{Teich}} \newcommand{\Comp}{\operatorname{Comp}} \newcommand{\Kah}{\operatorname{\sf Kah}} \newcommand{\Bir}{\operatorname{\sf Bir}} \newcommand{\Pos}{\operatorname{\sf Pos}} The work presented here (on the slides 13-15) {\bf \red is done in collaboration with Ekaterina Amerik.} \definition Let $M$ be a compact, K\"ahler manifold, $\Kah\subset H^{1,1}(M,\R)$ is K\"ahler cone, and $\overline\Kah$ its closure in $H^{1,1}(M,\R)$, called {\bf \blue the nef cone}. A {\bf\blue face} of a K\"ahler cone is an intersection of the boundary of $\overline\Kah$ and a hyperplane $V\subset H^{1,1}(M,\R)$ which has a non-empry interior. \conjecture {\bf \blue (Morrison-Kawamata cone conjecture)} \\ Let $M$ be a Calabi-Yau manifold. Then the group $\Aut(M)$ of biholomorphic automorphisms of $M$ acts on the set of faces of $\Kah$ {\bf \red with finite number of orbits.} \newpage {\bf \blue Birational K\"ahler cone} \remark Define {\bf \blue pseudo-isomorphism} $M\arrow M'$ as a birational map which is an isomorphism outside of codimension $\geq 2$ subsets of $M, M'$. \remark {\bf \purple For any pseudo-isomorphic manifolds $M,M'$, one has $H^2(M)=H^2(M')$.} \definition {\bf \blue Movable K\"ahler cone}, also known as {\bf \blue birational K\"ahler cone} and {\bf \blue birational nef cone} is a closure of a union of $\Kah(M')$ for all $M'$ pseudo-isomorphic to $M$. \conjecture {\bf \blue(Morrison-Kawamata birational cone conjecture)}\\ Let $M$ be a Calabi-Yau manifold. Then the group $\Bir(M)$ if birational automorphisms of $M$ {\bf \red acts on the set of faces of the movable cone with finite number of orbits.} \newpage {\bf \blue $(-2)$-classes on a K3 surface } \claim {\bf \blue (Hodge index theorem)} \\ Let $M$ be a K\"ahler surface. {\bf \red Then the form $\eta \arrow \int_M\eta\wedge\eta$ has signature $(+, -, -, ...)$ on $H^{1,1}(M,\R)$.} \definition {\bf \blue Positive cone} $\Pos(M)$ on a K\"ahler surface is the one of the two components of \[ \{v\in H^{1,1}(M,\R)\ \ |\ \ \int_M\eta\wedge\eta>0\} \] which contains a K\"ahler form. \definition A cohomology class $\eta\in H^2(M,\Z)$ on a K3 surface is called {\bf \blue $(-2)$-class} if $\int_M\eta\wedge\eta=-2$. \remark Let $M$ be a K3 surface, and $\eta\in H^{1,1}(M,\Z)$ a $(-2)$-class. {\bf \purple Then either $\eta$ or $-\eta$ is effective.} Indeed, $\chi(\eta) = 2+\frac{\eta^2}{2}=1$ by Riemann-Roch. \newpage {\bf \blue K\"ahler cone for a K3 surface} \theorem Let $M$ be a K3 surface, and $S$ the set of all effective $(-2)$-classes. {\bf \red Then $\Kah(M)$ is the set of all $v\in \Pos(M)$ such that $\langle v, s\rangle >0$ for all $s\in S$.} \definition {\bf \blue A Weyl chamber} on a K3 surface is a connected component of $\Pos(M)\backslash S^\bot$, where $S^{\bot}$ is a union of all planes $s^\bot$ for all (-2)-classes $s\in S$. {\bf \blue The reflection group} of a K3 surface is a group $W$ generated by reflections with respect to all $s\in S$. \remark Clearly, a Weyl chamber is a fundamental domain of $W$, and $W$ acts transitively on the set of all Weyl chambers. Moreover, {\bf \purple the K\"ahler cone of $M$ is one of its Weyl chambers.} \newpage {\bf \blue Cone conjecture for a K3 surface} \theorem Let $M$ be a K3 surface. {\bf \red Then $\Aut(M)$ is the group of all isometries of $H^{1,1}(M,\Z)$ preserving the K\"ahler chamber.} \corollary {\bf \red Morrison-Kawamata cone conjecture holds for a K3 surface.} {\bf \green Proof. Step 1:} A group $\Gamma$ of isometries of a lattice $\Lambda$ acts with finitely many orbits on the set $\{l \in \Lambda\ \ |\ \ l^2 = x\}$ for any given $x$ (see Kneser, {\em Quadratische Formen}, Satz 30.2). {\bf \purple Therefore, $\Gamma$ acts with finitely many orbits on the set of $(-2)$-vectors in $\Lambda$.} {\bf \green Step 2:} For each pair of faces $F, F'$ of a K\"ahler cone and $w\in O(\Lambda)$ mapping $F$ to $F'$, $w$ maps $\Kah$ to itself or to an adjoint Weyl chamber $K'$. Then $K'=r(K)$, where $r$ is the reflection fixing $F'$. In the first case, $w\in \Aut(M)$. In the second case, $rw$ maps $F$ to $F'$ and maps $\Kah$ to itself, hence $rw\in \Aut(M)$. \endproof \newpage {\bf \blue Hyperk\"ahler manifolds} \definition A {\bf \blue hyperk\"ahler structure} on a manifold $M$ is a Riemannian structure $g$ and a triple of complex structures $I,J,K$, satisfying quaternionic relations $I\circ J = - J \circ I =K$, such that $g$ is K\"ahler for $I,J,K$. \remark A hyperk\"ahler manifold {\bf \purple has three symplectic forms\\ $\omega_I:= g(I\cdot, \cdot)$, $\omega_J:= g(J\cdot, \cdot)$, $\omega_K:= g(K\cdot, \cdot)$.} {\bf\green REMARK:} This is equivalent to $\nabla I=\nabla J = \nabla K=0$: the parallel translation along the connection preserves $I, J,K$. {\bf\green DEFINITION:} Let $M$ be a Riemannian manifold, $x\in M$ a point. The subgroup of $GL(T_xM)$ generated by parallel translations (along all paths) is called {\bf \blue the holonomy group} of $M$. {\bf\green REMARK:} {\bf \purple A hyperk\"ahler manifold can be defined as a manifold which has holonomy in $Sp(n)$ } (the group of all endomorphisms preserving $I,J,K$). \newpage {\bf \blue Holomorphically symplectic manifolds} {\bf\green DEFINITION:} A holomorphically symplectic manifold is a complex manifold equipped with non-degenerate, holomorphic $(2,0)$-form. \remark Hyperk\"ahler manifolds are holomorphically symplectic. Indeed, $\Omega:=\omega_J+\1\omega_K$ is a holomorphic symplectic form on $(M,I)$. {\bf\green THEOREM:} (Calabi-Yau) A compact, K\"ahler, holomorphically symplectic manifold admits a unique hyperk\"ahler metric in any K\"ahler class. {\bf\green DEFINITION:} For the rest of this talk, {\bf \red a hyperk\"ahler manifold is a compact, K\"ahler, holomorphically symplectic manifold.} {\bf\green DEFINITION:} A hyperk\"ahler manifold $M$ is called {\bf \blue simple} if $\pi_1(M)=0$, $H^{2,0}(M)=\C$. {\bf \purple Bogomolov's decomposition:} Any hyperk\"ahler manifold admits a finite covering which is a product of a torus and several simple hyperk\"ahler manifolds. {\bf \red Further on, all hyperk\"ahler manifolds are assumed to be simple}. \newpage {\bf \blue The Bogomolov-Beauville-Fujiki form} {\bf \green THEOREM:} (Fujiki). Let $\eta\in H^2(M)$, and $\dim M=2n$, where $M$ is hyperk\"ahler. Then $\int_M \eta^{2n}=cq(\eta,\eta)^n$, for some primitive integer quadratic form $q$ on $H^2(M,\Z)$, and $c>0$ a rational number. {\bf \green Definition:} This form is called {\bf \blue Bogomolov-Beauville-Fujiki form}. {\bf \purple It is defined by the Fujiki's relation uniquely, up to a sign.} The sign is determined from the following formula (Bogomolov, Beauville) \begin{align*} \lambda q(\eta,\eta) &= \int_X \eta\wedge\eta \wedge \Omega^{n-1} \wedge \bar \Omega^{n-1} -\\ &-\frac {n-1}{n}\left(\int_X \eta \wedge \Omega^{n-1}\wedge \bar \Omega^{n}\right) \left(\int_X \eta \wedge \Omega^{n}\wedge \bar \Omega^{n-1}\right) \end{align*} where $\Omega$ is the holomorphic symplectic form, and $\lambda>0$. {\bf \green Remark:} {\bf \red $q$ has signature $(3,b_2-3)$.} It is negative definite on primitive forms, and positive definite on $\langle \Omega, \bar \Omega, \omega\rangle$, where $\omega$ is a K\"ahler form. \newpage {\bf \blue Birational nef cone on a hyperk\"ahler manifold} \definition A cohomology class $\nu\in H^{1,1}(M)$ is called {\bf \blue pseudoeffective} if it can be represented by a positive, closed current. \theorem {\bf \blue (Huybrechts, Boucksom)}\\ On any hyperk\"ahler manifold $M$, {\bf \red birational K\"ahler cone is dual (with respect to the BBF pairing) to the pseudoeffective cone.} Moreover, the birational K\"ahler cone is a union of K\"ahler cones {\bf \red for all hyperk\"ahler manifolds $M'$ pseudo-isomorphic to $M$.} \theorem {\bf \blue Divisorial Zariski decomposition (Boucksom). }\\ {\bf \red For any pseudoeffective $\nu$, we have $\nu = \nu_0 + \sum \alpha_i E_i$,} where $\nu_0$ is birationally nef, $\alpha_i$ positive numbers, and $E_i$ are exceptional divisors. \corollary Let $\eta\in \Pos(M)$ be an element of a positive cone on a hyperk\"ahler manifold. {\bf \purple Then $\eta$ is birationally nef if and only if $q(\eta, E)\geq 0$ for any exceptional divisor $E$.} \remark In other words, {\bf \red the faces of birational K\"ahler cone are dual to the classes of exceptional divisors.} \newpage {\bf \blue Divisorial reflections and Weyl chambers} \definition {\bf \blue Monodromy group} $\Gamma$ of a hyperk\"ahler manifold is a subgroup of $O(H^2(M,\Z),q)$ generated by monodromy operators for all Gauss-Manin local system associated with complex deformations of $M$. \remark The {\bf \purple monodromy group of a hyperk\"ahler manifold is a finite index subgroup in $O(H^2(M,\Z),q)$} (follows from global Torelli). \theorem {\bf \blue (Markman)}\\ For each exceptional divisor $E$ on a hyperk\"ahler manifold, {\bf \red there exists a reflection $r_E\in O(H^2(M,\Z))$ in the monodromy group fixing $E^\bot$.} \definition Such a reflection is called {\bf \blue a divisorial reflection}. \definition {\bf \blue Weyl chamber} on a hyperk\"ahler manifold is a connected component of $\Pos(M)\backslash E^\bot$, where $E^{\bot}$ is a union of all planes $e^\bot$ for all exceptional divisors $e$. \remark A Weyl chamber is a fundamental domain of a group generated by divisorial reflections. {\bf \purple Birational K\"ahler cone is one of the Weyl chambers.} \newpage {\bf \blue The proof of birational cone conjecture} \theorem Let $M$ be a hyperk\"ahler manifold. {\bf \red Then $\Bir(M)$ is the group of all $\gamma\in \Gamma$ preserving the birational K\"ahler chamber $\Kah_B$.} Here $\Gamma$ is the monodromy group.\\ {\bf \green Proof:} Follows from global Torelli. \endproof \corollary {\bf \blue (Markman)} {\bf \red Birational Morrison-Kawamata cone conjecture holds for hyperk\"ahler manifolds.} {\bf \green Proof. Step 1:} Let $\delta$ be the discriminant of a lattice $H^2(M,\Z)$, and $E$ an exceptional divisor. {\bf \purple Then $|E^2|\leq 2\delta$.} Indeed, otherwise the reflection $x \arrow x - 2\frac{q(x,E)}{q(E,E)}E$ would not be integer. {\bf \green Step 2:} A group of isometries of a lattice $\Lambda$ acts with finitely many orbits on the set $\{l \in \Lambda\ \ |\ \ l^2 = x\}$ for any given $x$ (see Kneser, {\em Quadratische Formen}, Satz 30.2). Therefore, {\bf \purple $\Gamma$ acts with finitely many orbits on the set of classes of exceptional divisors.} {\bf \green Step 3:} For each pair of faces $F, F'$ of a birational K\"ahler cone and $w\in O(\Lambda)$ mapping $F$ to $F'$, $w$ maps $\Kah_B$ to itself or to an adjoint Weyl chamber $K'$. Then $K'=r(K)$, where $r$ is the reflection fixing $F'$. In the first case, $w\in \Aut(M)$. In the second case, $rw$ maps $F$ to $F'$ and maps $\Kah_B$ to itself, hence $rw\in \Aut(M)$. \endproof \newpage {\bf \blue MBM classes} \definition {\bf \blue Negative class} on a hyperk\"ahler manifold is $\eta\in H^2(M,\R)$ satisfying $q(\eta,\eta)<0$. \definition Let $(M,I)$ be a hyperk\"ahler manifold. A rational homology class $z\in H_{1,1}(M,I)$ is called {\bf \blue minimal} if for any $\Q$-effective homology classes $z_1, z_2\in H_{1,1}(M,I)$ satisfying $z_1+z_2=z$, the classes $z_1, z_2$ are proportional. A negative rational homology class $z\in H_{1,1}(M,I)$ is called {\bf\blue monodromy birationally minimal} (MBM) if $\gamma(z)$ is minimal and $\Q$-effective for one of birational models $(M,I')$ of $(M,I)$, where $\gamma\in O(H^2(M))$ is an element of the monodromy group of $(M,I)$. \definition Let $(M,I)$ be a hyperkaehler manifold. A negative rational class $z\in H^{1,1}_\Q(M,I)$ is called {\bf\blue divisorial} if $z=\lambda [D]$ for some divisor $D$ and $\lambda\in \Q$. These properties are {\bf \purple deformationally invariant}. \theorem Let $z\in H^2(M,\Z)$ be negative, and $I, I'$ complex structures in the same deformation class, such that $\eta$ is of type (1,1) with respect to $I$ and $I'$. Then \\ {\bf \red {\ }\ \ \ * $\eta$ is divisorial in $(M,I)$ $\Leftrightarrow$ it is divisorial in $(M,I')$.\\ \bf \red {\ }\ \ \ * $\eta$ is MBM in $(M,I)$ $\Leftrightarrow$ it is MBM in $(M,I')$}. \newpage {\bf \blue MBM classes for $\Pic(M)=\Z$} The MBM and divisorial classes are better understood if the Picard group has rank one and generated by a negative vector (in this case $M$ is non-algebraic). \theorem Let $(M,I)$ be a hyperk\"ahler manifold, $\rk \Pic(M,I)=1$, and $z\in H_{1,1}(M,I)$ a non-zero negative class. {\bf \red Then $z$ is monodromy birationally minimal if and only if $\pm z$ is $\Q$-effective.} {\bf \green Proof. Step 1:} If $(M,I)$ has a rational curve, it is by definition minimal, hence represents an MBM class. {\bf \green Step 2:} If $(M,I)$ has no rational curves, it has no exceptional divisors, hence {\bf \purple birational K\"ahler cone is equal to the positive cone} (Boucksom). {\bf \green Step 3:} If $(M,I)$ has no rational curves, any pseudo-isomorphism from $(M,I)$ to another hyperk\"ahler manifold must be trivial. Indeed, pseudo-isomorphisms are birational, and exceptional locus of a birational map is covered by rational curves. Then $\Kah_B=\Kah$: {\bf \red birational K\"ahler cone is K\"ahler cone is positive cone.} \endproof \remark This argument proves that {\bf \purple MBM classes correspond to faces of a K\"ahler cone} for $\rk \Pic(M,I)=1$. \newpage {\bf \blue MBM classes and the K\"ahler cone} \theorem Let $(M,I)$ be a hyperk\"ahler manifold, and $S\subset H_{1,1}(M,I)$ the set of all MBM classes in $H_{1,1}(M,I)$. Consider the corresponding set of hyperplanes $S^\bot:=\{W=z^\bot\ \ |\ \ z\in S\}$ in $H^{1,1}(M,I)$. {\bf \red Then the K\"ahler cone of $(M,I)$ is a connected component of $\Pos(M,I)\backslash \cup S^\bot$}, where $\Pos(M,I)$ is a positive cone of $(M,I)$. Moreover, for any connected component $K$ of $\Pos(M,I)\backslash \cup S^\bot$, there exists $\gamma\in O(H^2(M))$ in a monodromy group of $M$, and a hyperk\"ahler manifold $(M,I')$ birationally equivalent to $(M,I)$, such that $\gamma(K)$ is a K\"ahler cone of $(M,I')$. \remark {\bf \purple MBM classes correspond to faces of the K\"ahler cone.} \remark {\bf \red Morrison-Kawamata cone conjecture would follow if we prove that monodromy group acts on the set of MBM rays with finitely many orbits.} This is implied by the following conjecture. \conjecture Let $M$ be a hyperk\"ahler manifold. {\bf \purple Then there exists a constant $C>0$, such that for any minimal rational curve $S\subset (M,I)$ with $q(S,S)<0$, one has $|q(S,S)|